Advanced Calculus Single Variable

2.6 The Binomial Theorem

Consider the following problem: You have the integers Sn =

{1,2,⋅⋅⋅,n }

and k is an integer
no larger than n. How many ways are there to fill k slots with these integers starting from left
to right if whenever an integer from Sn has been used, it cannot be re used in any succeeding
slot?

◜----k of the◞s◟e slots-◝
--,---,---,--,⋅⋅⋅,---

This number is known as permutations of n things taken k at a time and is denoted by
P

(n,k)

. It is easy to figure it out. There are n choices for the first slot. For each choice for the
fist slot, there remain n − 1 choices for the second slot. Thus there are n

(n − 1)

ways to fill
the first two slots. Now there remain n − 2 ways to fill the third. Thus there are
n

(n − 1)

(n − 2)

ways to fill the first three slots. Continuing this way, you see there
are

P (n,k) = n (n− 1)(n− 2)⋅⋅⋅(n− k+ 1)

ways to do this.

Now define for k a positive integer,

k! ≡ k (k − 1)(k − 2)⋅⋅⋅1,0! ≡ 1.

This is called k factorial. Thus P

(k,k)

= k! and you should verify that

P (n,k) =---n!--
(n− k)!

Now consider the number of ways of selecting a set of k different numbers from Sn. For each
set of k numbers there are P

(k,k)

= k! ways of listing these numbers in order. Therefore,
denoting by

( n )
k

the number of ways of selecting a set of k numbers from Sn, it must be
the case that

( )
n ---n!--
k k! = P (n,k) = (n − k)!

Therefore,

( n ) n!
k = --------.
k!(n − k)!

How many ways are there to select no numbers from Sn? Obviously one way. Note the above
formula gives the right answer in this case as well as in all other cases due to the definition
which says 0! = 1.

Now consider the problem of writing a formula for

(x+ y)

n where n is a positive integer.
Imagine writing it like this:

n times
◜---------◞◟--------◝
(x + y)(x + y)⋅⋅⋅(x + y)

Then you know the result will be sums of terms of the form akxkyn−k. What is
ak? In other words, how many ways can you pick x from k of the factors above
and y from the other n − k. There are n factors so the number of ways to do it
is

( n )
k .

Therefore, ak is the above formula and so this proves the following important theorem known
as the binomial theorem.

Theorem 2.6.1The following formula holds for any n a positive integer.